On stability of probability laws with univariate stable marginals

  • Evarist Giné
  • Marjorie G. Hahn
Article

Summary

Examples of D. Marcus in ℝ2 dispel the belief that a probability measure on ℝd is stable if and only if all its univariate marginals are stable. However, in ℝd (in fact, in fairly general linear spaces), a probability measure whose two-dimensional marginals are all infinitely divisible is stable if and only if all its univariate marginals are stable.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Araujo, A., Giné, E.: The Central Limit Theorem for Real and Banach Valued Random Variables. New York: Wiley 1980Google Scholar
  2. 2.
    Dudley, R.M., Kanter, M.: Zero-one laws for stable measures. Proc. Amer. Math. Soc.45, 245–252 (1974)Google Scholar
  3. 3.
    Ferguson, T.: On the existence of linear regression in linear structural relations. U. California Publ. in Statistics.2, N∘ 7, 143–166 (1955)Google Scholar
  4. 4.
    Ferguson, T.: A representation of the symmetric bivariate Cauchy distribution. Ann. Math. Statist.33, 1256–1266 (1962)Google Scholar
  5. 5.
    Hahn, M.G., Hahn, P., Klass, M.J.: Pointwise translation of the Radon transform and the general central limit theorem. Ann. Probability11, 277–301 (1983)Google Scholar
  6. 6.
    Lévy, P.: The arithmetical character of the Wishart distribution. Proc. Cambridge Phil. Soc.44, 295–297 (1948)Google Scholar
  7. 7.
    Lévy, P.: Théorie de l'Addition des Variables Aleatoires. 2nd edition. Paris: Gauthier-Villars 1954Google Scholar
  8. 8.
    Linnik, J.V., Ostrovskii, I.V.: Decomposition of random variables and vectors. Transl. of Math. Monographs 48. Providence, R.I.: Amer. Math. Soc. Publ. 1977Google Scholar
  9. 9.
    Marcus, D.: Non-stable laws with all projections stable. Private communication 1982. Z. Wahrscheinlichkeitstheorie verw. Gebiete64, 139–156 (1983)Google Scholar
  10. 10.
    Philipp, W.: Weak andL p invariance principles for sums ofB-valued random variables. Ann. Probability8, 68–82 (1980)Google Scholar
  11. 11.
    Press, S.J.: Applied Multivariate Analysis. New York: Holt, Rinehart and Winston 1972Google Scholar
  12. 12.
    Press, S.J.: Stable distributions: probability, inference, and applications in finance: a survey and a review of recent results: In “Statistical Distributions in Scientific Work”, ed. by G.P. Patil, S. Kotz and J.K. Ord. Dordrecht, Holland: D. Reidel Publ. Co. 1975Google Scholar
  13. 13.
    Schaefer, H.: Topological Vector Spaces. Berlin, Heidelberg, New York: Springer 1971Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • Evarist Giné
    • 1
  • Marjorie G. Hahn
    • 2
  1. 1.Department of MathematicsLouisiana State UniversityBaton RougeUSA
  2. 2.Department of MathematicsTufts UniversityMedfordUSA

Personalised recommendations